If you change the way you are organizing your information a bit, then you could make use of the Checklist feature that each card has. This would allow you to easily check-off items as they are completed.

You can do this several ways, depending how you want to organize your lists and flow. There is no "this card is completed" function per se, it should be defined in a different way. The first (seperate list) is my favorite, but the second works as well for some projects. I wouldn't personally use my third option, but it is a valid method I guess.

'standard' left-to-right flow: Make a ToDo and Done list or even a ToDo, Doing and Done list. Move the cards from left to right. Drag them with your mouse or use < and >

Archive all done tasks, via the card's menu or using c

You can use labels. No label for 'random' cards, and a label for done. Maybe also a label for 'doing'. Use the menu on a card, or 1, 2 etc keyboard shortcuts

There are probably more things you can think of, like assigning todo's to yourself and removing the assignement if it is done, or the other way around, assigning yourself if you have done it, although that would mess up your 'cards' view a bit with completed tasks.